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Zbl 0842.45009
Lizama, Carlos
Uniform continuity and compactness for resolvent families of operators. (English)
[J] Acta Appl. Math. 38, No.2, 131-138 (1995). ISSN 0167-8019; ISSN 1572-9036/e

The author studies the following Volterra convolution equation $$V_A u(t):= u(t)- \int^t_0 k(t- s)Au (s) ds= f(t), \qquad t\in J:= [0,T ]$$ where $X$ is a Banach space, $A$ a densely defined closed operator on $X$, and $k\in L^1_{\ell x} (\bbfR_+)$. A strongly continuous family $\{R(t) \}_{t\geq 0}$ of bounded operators on $X$, commuting with $A$, is called a resolvent family for the above equation if $V_A Rx= x$ for all $x$ in the domain of definition ${\cal D} (A)$ of $A$. Conditions for the existence of $R(T)$, their regularity, positivity and other properties of $R(t)$ are widely studied. \par In the present work the uniform continuity of $\{R(t) \}_{t\geq 0}$ and compactness of $R(t)- I$ for $t>0$ are characterized in terms of operator $A$ provided the kernel $k$ is regular. In particular cases $k(t) \equiv 1$, $k(t) \equiv t$ these results coincide with the known ones on $C_0$-semigroups and cosine-families of operators, obtained by J. Guthbert, H. Henriques, D. Lutz, J. Pruss and others.
[R.Duduchava (Tbilisi)]

MSC 2000:: *45N05 Integral equations in abstract spaces
45G10 Nonsingular nonlinear integral equations
47D09 Operator cosine functions and higher-order Cauchy problems
47D06 One-parameter semigroups and linear evolution equations

Keywords: semigroup of operators; Volterra convolution equation; Banach space; resolvent family; uniform continuity; compactness; cosine-families of operators

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